Fellow Mathers, I am currently in a Euclidean Geometry class. We are using the SMSG axioms to build our geometry. I am struggling in proving the fact that all triangles can be inscribed into a circle, i.e. that...

$Ɐ$ triangles $ΔABC$, $∃$ a circle $Θ$ such that $A,B,C ∈ Θ$.

Anyone who can provide a proof or point me in the right direction, it would be greatly appreciated. - SDH

  • $\begingroup$ This question may help: math.stackexchange.com/questions/16634/… $\endgroup$ – David Peterson Sep 15 at 4:55
  • $\begingroup$ In the title "cyclic" is misleading. Periphrasing into "can be inscribed in a circle" would be probably better. $\endgroup$ – Jean Marie Sep 15 at 5:49
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    $\begingroup$ @JeanMarie Will do. Don’t really know what I’m talking about with the jargon, just saw that terms being used online. $\endgroup$ – SDH Sep 15 at 5:51

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Let be $D$ the intersection point of the perpendicular bisectors of $AC$ and $BC$. You have that

  1. $AD=CD$ since $D$ belongs to $s$
  2. $CD=BD$ since $D$ belongs to $r$ Therefore $AD=BD=CD=R $ and the circle of center $D$ and radius $R$ is the required circle.

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